I am numerically studying a class of matrices $A$, size $n \times n$, that are singular, but can be decomposed into $$A = V D V^T,$$ where $D$ is a (non-singular) square, diagonal matrix of size $m \times m$ and $V$ is a matrix of size $n \times m$. Does this decomposition have a name?
I am trying to solve for $$ \arg \min_x ||A x - b||^2.$$ My question is: is the known decomposition useful in any way? Had it been an SVD decomposition, it would immediately give the answer to the equation.
In general $n < m$, and an example of $V$ could be
V = [[-1, -1, 0, -1, 0, 0, 0, 0],
[ 1, 0, -1, 0, 0, -1, 0, 0],
[ 0, 1, 1, 0, -1, 0, -1, 0],
[ 0, 0, 0, 1, 1, 0, 0, -1],
[ 0, 0, 0, 0, 0, 1, 1, 1]]
$D$ can then be any diagonal matrix, but with the identity matrix, we have
A = [[ 3, -1, -1, -1, 0],
[-1, 3, -1, 0, -1],
[-1, -1, 4, -1, -1],
[-1, 0, -1, 3, -1],
[ 0, -1, -1, -1, 3]]
which has rank $n - 1$.